First, I would NOT request a death certificate until you have more information. Why spend money for a search when he might not be there? A DC in that time period may not tell where he was buried.
I found reference to your "private" family tree so help there wasn't available. I had to access the WWI Registration to identify when he was born, then the 1910 census to find out where. When you post it is helpful for volunteers to identify these criteria pierces of information. There were MANY Pearl Rhea's, one has to get the criteria before additional can be researched.
The WWI Draft Registration
Pearl E. Rhea, born May 27, 1874 (US Citizen), living at Casper, WY. Boilermaker for the Standard Oil Co. Nearest Relative Mrs Elizabeth Rhea. He registered on 9/12/1918, so unlikely he served.
The 1910 Census of Lancaster, Nebraska
Pearl E Rhea, age 35, born in Nebraska to both parents born in Illinois, Fireman/Rail Road
Lizzie, age 34, born in NE to both parents born in Wales/Welsh speakers, mother of 4/3 living.
Because of his job types in 1910 and WWI Draft, it doesn't appear likely he would suddenly begin farming.
I found one posted family tree containing Pearl, son of David V. Rhea 1836-1903 and Mary Rosina "Rose" Judd, 1831 to 16 Jul 1930 in Glendale, Los Angeles Co, CA. Her obit "might" identify Pearl as preceding her but not enough chance to identify where. This tree showed Pearl with background as a female. http://www.rootsweb.ancestry.com/~wynatron/cemeteries.htm
identifies some cemetery records in Natrona County. But no transcriptions. http://www.rootsweb.ancestry.com/~wynatron/Obits.htm
This might be your next logical step...find out if he died there.